(2) Prove that the following are equivalent for x ER and A CR. (a) X E...
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n > 0, ELY?"] = 1207) and E[x2n+] = 0.
3. Let S CR and a ER, and define aS = {as : SES}. (a) Show that if a > 0, then sup aS = a sup S. (b) Find an example where sup as a sup S.
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
Evaluate the following integrals. (a) / In(3x) dx for x > 0 (e) / ( +er) dx (n lete* dx (e) sin(5x +1) dx
Prove this using the definition R7: log(n*) is O(log n) for any fixed x > 0
| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.
Prove that is an integer for all n > 0.
Use induction and Pascal's identity to prove that (7) = 2" where n > 0.
Prove by mathematical induction (discrete mathematics) n? - 2*n-1 > 0 n> 3